In spite of many interesting attempts, the problem of automatically finding alignments in a 2D set of points seems to be still open. The difficulty of the problem is illustrated here by very simple examples. We then propose an elaborate solution. We show that a correct alignment detection depends on not less than four interlaced criteria, namely the amount of masking in texture, the relative bilateral local density of the alignment, its internal regularity, and finally a redundancy reduction step. Extending tools of the a contrario detection theory, we show that all of these detection criteria can be naturally embedded in a single probabilistic a contrario model with a single user parameter, the number of false alarms. Our contribution to the a contrario theory is the use of sophisticated conditional events on random point sets, for which expectation we nevertheless find easy bounds. By these bounds the mathematical consistency of our detection model receives a simple proof. Our final algorithm also includes a new formulation of the exclusion principle in Gestalt theory to avoid redundant detections. Aiming at reproducibility, a source code and an online demo open to any data point set are provided. The method is carefully compared to three state-of-the-art algorithms and an application to real data is discussed. Limitations of the final method are also illustrated and explained.